This

one.

So

if.

Thank you.

OK.

And we go.

Thank you.

It's always a mistake to
detach the microphone and

give it to the person who introduces you.

Thank you to a guy for the introduction.

It's a real pleasure to be here.

Back at Georgia Tech.

I used to visit quite often in
the early one nine hundred ninety S.

late one nine hundred eighty S. and
one of the real pleasures was running into

John Ford or having dinner with Joe
talk I said he was a real character.

Certainly a very very
perceptive scientist and.

I do feel it's a real privilege to be
here given a seminar in his honor.

Now the title of the told
is a little strange because

I originally sent target I.

Our normal title you might
say any mediately said.

Back to me and
said This isn't sexy enough.

It's not until forward.

Enough.

And so I came up with this title by
thinking about some of the famous

Joe Ford quotations.

For example Joe said God does play
dice for the dice are loaded.

Joe obviously was it was
a an expert in a pioneer in

not just chaos but
also the field of quantum calles.

And Joel concluded from
his studies of the.

There are no current mop.

A classical math in that in fact
quantum mechanics is intrinsically

incomplete because in fact when you
look at a classical system which is

chaotic then it is impossible to reproduce
that behavior use in quantum mechanics.

Now I don't work in quantum calles I work
in classical chaos and quantum mechanics

so my talk is not actually about quantum
calles but it is comes in two parts.

It's about applications of chaos theory
in understanding the structure of

the bells and in particular the formation
of binary objects in the Kuyper belt and

also it's about quantum mechanics so
it's not quantum chaos it's chaos and

want to mechanics is essentially
two separate subjects.

So I can't do justice to quantum
chaos in the same way that Joe would.

So I decided to organize
the seminar around.

Joe Ford quotations for
which he was very famous and

well known here is the one I alluded to
a region where you are God does play dice.

The interesting thing about this
school Taishan is that he says.

God plays dice with the dice are loaded
and the main objective of physics is to

find out by what rules they
were loaded and I think this.

Joel up because he had a real
curiosity about how nature works and

he wasn't satisfied with just
coming up with an answer.

He really wanted to get to the bottom
of this very philosophical questions.

Determinism calles in the solar
system chaos in the universe.

This quotation I like very much.

He says thus far we have cascade else
in the traditional role as villain and

certainly when we think of chaos.

We think of it in a relatively
negative sense because naturally we're

inclined to to look for order and there's
a feeling that if something is chaotic.

It really is destructive.

It is it is a villain and so

Joel talked to Kelsey's be in that cast
in the role of a villain but he said.

Yet when controlled the villain
the scales becomes gentle.

Useful even and chanting.

Chaos can therefore provide us with
a virtual soul display of exciting variety

a richness of choice a core new Copia of
opportunity this is classic Joe Ford.

I mean I can almost hear him say I'm sure.

So I can utter in these words.

But that there is more to this than just.

Colton Joe because Joe recognized that
chaos provides a lot of flexibility for

us to control systems and
one of the things that I want to try

to get across to him I told is in fact
chaos in the solar system is left.

It's fingerprint on various
aspects of the the solar system in

particular aspects having
to do with the belt.

Now.

Rather than speak for himself.

So how

do I do I have just a.

Told you.

Yeah.

Which is this.

I know you're taking a break.

So we need to adjust the volume here.

OK.

So what it is.

So this is this is an amazing.

Into which language.

Google lease.

OK OK.

So Joe says you know calles
his from another planet and

he's the ordained if I'm
journalist of of chaos and so

what I want to talk to you about
today is chaos in the solar system.

And in particular how what Joel spoke
of his being in the Oneness chaos

in fact him be an organizing principle in
the formation of certain structures in.

In our solar system.

Now the Kuyper belt is often referred
to as the the from Tier the third

part of the solar system it's way
out beyond Neptune and Pluto which

obviously is no longer a planet Pluto
discovered in one nine hundred thirty.

Then there was a hiatus until about one
nine hundred seventy eight when Sharon

which is Pluto is moon
was was discovered and

since that time with with
the Hubble Space Telescope and

ground based astronomy something
like seventy thousand or

more objects have been found in the
transit union part of the solar system.

Well so what I mean these things
just rocks orbit in the sun.

What's the big deal.

Well it turns out that something
like ten percent of these

objects in the belt
are actually binary is.

Now binary is very important in astronomy
much of what we know about stars is

because most stars in the galaxy in
the universe are actually binary.

And if you know that an object is
a binary you can figure out these orbit.

You can figure out the total mass and
so a lot of.

Information is is extractable
by looking at binary objects so

it's a real godsend in a sense that
the Kuyper belt contains so many binary is

the camper belt is thought to be
a primordial remnant of the solar system.

That is essentially pristine.

So the objects in the binary
in the in the Kuyper belt.

Tell us about conditions in
the early solar system and

they also provide sort of
a next very mental test bed for

thinking about theories of how
the solar system originated.

Now at one time people thought that
our solar system was typical but

with all of the planets that have been.

It's covered over the past few years
it's becoming clear that solar

systems come in many many
different flavors and in fact.

Ours is is quite remarkable.

And so any insight that we can get
into the origin of the solar system

is obviously very important.

Now the finding in the something like
you know ten percent of the objects in

the Kiper belt binary is
is is quite remarkable.

Here we're looking at the different
populations of the belt objects

as a function of distance from
the sun most of the binary is

found in this region just beyond
Pluto as objects be on Pluto's orbit

and so the classical part of the Kuyper
belt is here at about forty to

forty five astronomical units from
the Sun There is also a region

where Pluto inhabits which is known as
the three to resonance in this region.

There is a resonance between Neptune and
objects in there in and Pluto.

So it turns out the the distribution
of binary is in the Kuyper belt

is quite different between
the classical region and

the three to resonance which is something
that I want to talk about a little later.

So here is a is a blowup of what was shown
before and what what is being illustrated

are the distribution of binary is
in the Kuyper belt is a function

of distance from the sun and also as
a function of their eccentricity So

the eccentricity is essentially
how cigar shaped the orbit is and

one of the interesting and
remarkable findings is that

many of the objects in the Kuyper
belt which are binary is

consist of bodies which
are similarly sized to each other.

So if we think about the earth the moon
for example as being a binary.

The earth is very very much
larger than it is is the moon and

in fact the center of mass of
the Earth moon system resides in the.

Actually in the earth.

But in distinction to this another
binary is in the solar system.

It turns out that in the Kiper belt
the binary partners are roughly

equal sized and
that provides a clue as to their origin.

So if we compare the.

Physical and

orbital properties of binary as in
the type of belt to other binary is in

the solar system we can actually see that
there are some quite striking differences.

So in particular if we have some
generic binary A is the semi-major axis

essentially the distance between the two
binary partners and we think about

the radius of the primary to the radius
of the secondary It turns out that in.

For example the main belt of
the asteroid belt the main belt or

near earth asteroids.

There is a big disparity in size
between the primary and the secondary.

And that suggests a collision
all origin for these things.

So if you have a collision between two
particles for example a planet and

the earth which is thought to
produce the moon then use that

there is quite a big science discrepancy
or disparity between the two objects.

If you look at the trams Neptunian
are the capable binary is

in fact the radius of the primary to the
radius of the second tree is quite large.

So these things are roughly equal sized
which implies that they had a known

coalitional origin.

So you can plot these things in you
know various ways but however you do it.

It turns out that capable of binary is
quite distinct from other population.

Binary as in the solar system.

It also turns out that the distance
between the particles which constitute

the binary a very large in comparison to
binary is which have a collision old.

Origin.

Another strike in feature of
by news in the belt is that

the eccentricity of the mutual orbit so
as these things go around the sun

the basically following a relatively
circular heliocentric orbit but

is the orbit each other
then the eccentricity is

quite large much larger than is is found
in other parts of the solar system.

So here is an example of one
such binary This is an actual

observation this is two thousand and
seventy why for thirty and

the this is between these two bodies
is on the order of twenty one thousand

kilometers so this is a very wide
binary given the fact that the objects

themselves are only on the order of around
one hundred kilometers in diameter.

So you have relatively small bodies
which are orbiting each other

at very very large distance.

OK So one of the issues
when when people discovered

a large fraction of binary is in
the old was what is the origin.

I would these things form and so
we decided to look at this problem fairly

simplistically by fits thinking
about having two body is and

the sun in the context of the circular
restricted three body problem.

So this is you know obviously one of
the oldest problems in in physics

where you have the sun a planet or

another particle orbiting the sun
on a circular orbit and then.

We have a third body.

The idea is that the the in the circular
restricted three body problem is that

the planet is confined
to a circular orbit and

it is unaffected by
the motion of the third body.

If you go to a non-inertial

frame that is rotating with the mean
motion of the planet around the sun

then the circular restricted
three body problem emerges.

So the idea is to think about the
formation of binary as in the caper belt

basically using the circular
restricted three body problem.

Now.

One of the difficulties in
dealing with this system is that.

There is no potential energy surface.

Because when you go to
the non-inertial frame

then it turns out that
you have a non conserved.

I get a momentum in the problem.

And a way to deal with that is
rather than writing down Tony and

directly in terms of my mentor if we
write the Hamiltonian down in terms of.

Velocities.

Then you come up with what is called
a surface of zero velocity so

the expression for the energy looks
like a quadratic form in philosophy as.

Plus this term here and
this surface looks like this.

So you have in this problem.

Several equilibrium points so
we have a warm L two and

L three which are subtle points.

And you also have two maxima.

Which are located here and here and
actually at these mikes similar reside

the Trojan asteroids which
are two groups of asteroids

that along the same orbit as
Jupiter in the solar system.

OK so here in this problem.

This serves as our effective potential
energy surface although it is not

a potential energy surface.

Strictly speaking because it is obtained
by setting the velocity is equal to zero.

Here is a three dimensional picture of

this region that resides between L one and
L two and this is called the hills fear.

So in the three body problem for
example the Earth Moon Sun system

the hill sphere is the region in which the
gravitational attraction of say the earth.

And the moon is more important
than the part about it is

more important than the perturbation from
the Sun So even though both the Earth and

the moon are orbiting the sun
within the hill sphere.

It is the local gravitational
attraction which is dominant but

the sun obviously cannot
be there collected and so

the sun can be treated as a part
of beige and inside this region.

So for example in the case of the earth
and the Moon the Moon is about.

On an orbit which is about thirty
percent of the Earth Moon Hill sphere.

OK so in trying to understand
the origin of binary objects in the.

Kuyper belt then any model has to explain.

The fact that the secondary and

primary members of the binary
have roughly equal masses.

It also has to explain the origin
of the moderate eccentricity So

the eccentricity of the mutual
orbit is neither very low or

very large it's somewhere in between and
also the very very large.

So the major axes of these systems.

So there are several
components are several.

Features of type of belt.

Binary which a formation
model has to us to explain.

OK so how do these things form
while they're been various.

Theories put forward including one for you
to write in feeling in two thousand and

two essentially the idea behind
this model is that you have.

A collision between two particles
within the hill sphere of a larger body

these two things come together collide and
marriage and form a binary.

There are other formation
models involved in.

Very similar mechanisms
where whereby two particles

become gravitationally entangled and
eventually form a binary.

Now one of the one of the issues is if you
have two two particles which come together

then clearly they have enough
energy to break up and so there has

to be some dissipative mechanism by which
these these binary can be stabilized.

My guess is.

OK so we were lucky in.

Capture mechanisms in the solar system and

in particular the role
of payoffs in capture.

So what this movie will show is
the structure of face to face

as a function of energy.

So here is a second through
the surface of zero velocity.

And this is what a plank or a surface
a section looks like at this energy so

what the movie will show is as
a function of energy how face to face

makes a transition from regular
motion to too chaotic motion.

So here the energy is increasing.

Now what you notice is that this
island here which corresponds to

retrograde orbits is still stable whereas

the other island which corresponds
to progress orbits has disappeared.

But the main thing to notice is that.

As a function of energy.

You have interspersed regions
of regular motion and Cayle So

here is regular motion but
right in this region.

We have a thin layer of Cale's which
will turn out to be important for

the formation of these objects.

So here is is how we can
visualize the face space in

the three body Hill problem or
in the restricted three body problem

essentially you have
three kinds of motion.

We have regular motion which is.

So this is a point or a sort of section
here we have a periodic orbit and

surrounding these periodic orbits.

We have as I periodic orbits are K.
am I so this is regular motion.

We also have a scattering reach him.

So if particles started out with initial
conditions in the scattering region.

Essentially they immediately
leave the hills fear.

But at the interface between regular and
chaotic motion.

We have a thin layer of Cale's.

Now as a function of energy the shape
of these islands changes but

nevertheless this structure itself is

robust over much of the energy
region of interest.

Namely that we have regular motion
separated from a very fast got free motion

by chaotic orbits.

Now what happens is that if a particle

becomes cult up in one of
these chaotic regions.

Then you can actually find form.

A transience binary in this case.

So here we have two particles
which are orbiting in the sun.

They've.

Scattered off each other but

they've done Sol such that they're
in one of these chaotic regions.

And these chaotic regions
are sticky in the sense that two

particles which come together.

It's like a resonance in atomic
physics will orbit each other for

a very long time.

But if you just wait long
enough then eventually

the particle will escape from the chaotic
layer and the binary will will break up.

Now these K K M regions
correspond to regular motion but

there is no way for two particles to
actually enter into the regular region.

So two particles that come together.

Become.

And meshed in this this chaotic layer and
they can live for

a very long time but
eventually the binary will break up.

So our idea was that in essence the.

Chaotic Lehrer's are the glue.

That allows a binary to form temporarily.

And during the time
the transience binary is

an existence then there can be some
other dissipative mechanism which sucks

energy out of the system and puts the
binary into one of these regular retunes.

If you are just in the very fast
scanner in parts of the hill sphere.

Then the binary can only live for
maybe ten or twenty years.

It's a very unstable SR But if you get
caught up in one of these chaotic regions.

Then in fact the binary can live for

obits of ten thousand years or
more in the copper belt.

So the idea is that you have

binary formation where the two particles
are caught up in a chaotic region.

And then there is another mechanism
in this case we propose that.

Scantron by another particle will actually

extract some energy and
Kohls the binary to become stable.

So here we have an example of a transients
binary orbit which is caught up in one

of these chaotic regions
another particle comes in and

scatters this particle may or may not
stabilize the binary in principle it

can also destabilize it but
a certain fraction of.

Intruder particles will
extract some energy and

push the binary from a chaotic
orbit to a regular orbit.

So here we're plot in the radius of
the two particles which are initially.

Part of a transients or a chaotic binary
the intruder particle comes in and

leaves and you see the transition
from regular motion to relatively

from chaotic motion to relatively
regular periodic motion.

Now Rachel we came interested in this
problem from the point of view of

chemical dynamics and
transition state theory.

So here.

What I'm showing is.

The two subtle points wall and L two
in the restricted three body problem.

And here we have particles which started
out in the unstable money fault of

the transition state and here particles in
the stable part of the transition state.

So these are trajectories whose
initial conditions are such that

when I run the movie.

These will as scape.

Rolls these particles will actually
enter into the hill sphere and

you can see that many of them survive for
very very long times.

So they immediately are scape.

So these initial conditions
are randomly chosen but

correspond to binary which
transiently formed and

which you know live for many many
periods inside of the whole sphere.

Now if we increase the energy
then in that case were high

above the subtle points then the behavior
is qualitatively the same but

even at this very high energy source
somewhere up here on the energy surface.

You can see that some initial conditions
correspond to binary which live for

a very very long time.

So again this started in the stable
part of the money for old.

And what's being shown are initial
conditions each one of these

initial conditions corresponds to
a binary sort of some of these

particles a living for
a very very long time sir.

Well we're going to come to that.

OK I'm going to.

I mean so right now I'm just
describing the simulations that we did.

So our hour of sort of.

STARTING POINT was that perhaps chaos is
the mechanism that extends the lifetime

of the of a binary sufficiently long
enough that it can be stabilized.

So we did the simulations I mean a fairly
straightforward calculation we identified

initial conditions which can actually
penetrate inside the hills here and

we found that the probability of capture
of a binary in fact depended very

much on the mass ratio which was
a completely unexpected finding.

So we didn't build this
into the simulation.

We simply.

Didn't want classical Monte Carlo
simulations in the circular restricted

three body problem and looked at
the probability of forming a binary

which was stabilized
very intruder scattering.

And here the probability is quite
different for if the binary partners

have equal masses as compared to the case
where they have very different masses.

So this was luck because
you know we didn't.

Build or

even expect to have any dependence upon
the mass of the of the binary partners.

And in fact in the whole problem.

The mass ratio of the two to two partners
can be scaled out of the problem.

So there is no dependence upon mass
ratio in the whole problem itself.

But when we take the whole problem and
fire intruders at the binary we find that.

If the prototype binary partners
have roughly equal amounts.

There is a significant improvement
in Capture probabilities.

So this was something that came up and
expectantly.

And so obviously we wanted to
understand you know what is the origin

of this effect.

So here I'm just showing
the result of our simulation.

Here is the eccentricity of
the stabilized binary versus the seven

major axis as a function of the as a
fraction of the radius of the hill sphere

the yellow points are for
cases where the binary partners have

roughly equal mass ratio or
equal mass ratio and the turquoise

forms are when there is a big difference
in mass between the two particles.

So we found that.

The eccentricity is that of
predicted by this model namely

OK also assisted capture with the chaos.

I was transients binary to form which is
then stabilized certainly comparable to

observation and to be more observations
since then we find moderate eccentricity

is an Also we find semi major axes
which are in the in the observe range.

But then the question that was just
announced is what is responsible for

the mass ratio effect.

Why is it that binary
partners which have roughly

the same mass stabilized preferentially.

Well.

So are scenario is that we have a quasar
you've been around binary which is formed

and then include a particle
sconces from the system and

may extract some energy there
by stabilizing the system so

we look to the residence time for Include
of particles inside the hills fear.

And it turns out that in
the scattering region of phase space.

Intruders.

Scatter into and
out of the whole sphere very quickly.

So there is a natural separation
in time scale between

the mutual orbital period
of the transience binary.

And an intruder particle
which is scattered from it.

So the intruder comes in and
leaves very very quickly.

So we made an approximation and
A about it.

Separation of time scales and
said since the intruder

particle spends relatively little
time inside of the hill sphere.

Let's throw out the sun and
assume the the but

I'm sorry orbit even
though in the long time.

It is chaotic because it is caught up
in one of these regions very close

to a regular part of
free space overworn or

two periods the themes to follow
are relatively periodic orbits.

So by throwing out
the sun we can now reduce

this problem to the elliptic
restricted three body problem.

Elliptic because the binary
partners in general

won't be following a circular orbit
will be following an elliptical one.

And so just so here's the intruder we
just sort of comes in does its thing and

leaves either stabilizes or
destabilize a system but

it's there for
a relatively short period of time.

Now in study of the elliptic restricted
three body problem we can no longer

use plunker a surface of
section because the the humble

tone is explicitly dependent upon time so
we can't we can't use.

Poincaré serves as a section so
we we use something called the fast enough

indicator which is one of the many calles
indicators which people developed.

So here it would take in
a conventional crank or

a sort of piece of section in the circular
restricted three body problem and just.

There's a calibration We're comparing
the map that the fastly up and

off indicator generates for
the same system and

you can see there's a very nice
correspondence between regular and

chaotic regions and
it's the chaotic see that is

surrounding this regular region which
is important in in this mechanism.

And I notice that the the this
is like an island the beach

is essentially chaotic but
then if you're outside of this region.

You have fast scattering So these
are the three different parts of a space.

And we love to turn in this model
this idiotic model we learnt

how the structure of face to
face changes as a function of.

The eccentricity of the binary orbits and

also as a function of the mass
ratio of the two partners.

And it turns out that if you if you
take mass ratios which are close to

one as a here and
here and mass very shells

which are quite different from each other
then face space looks very different and

so this is where we postulated that
the Mass Effect is coming from namely

how effective is the intruder
stabilize in the transition vinery.

Now if you look at this one the mass
ratio is one the eccentricity is zero.

So the binary is assumed to be
following a circular orbit.

There's a lot of Cale's.

These yellow regions
correspond to chaos and

here we have regular regions and
so anybody Noory.

We live in in this region.

When an intruder approaches it.

The probability is that the intruder
will become caught up in this sea of

Cale's In other words it will spend
a lot of time inside the hills fear.

Because it takes a long time for these.

Chaotic orbits to exit
from the hills fear.

So if you think about trying to stabilize
one of these frog gyal binary systems.

The longer the intruder spends
in close proximity to the binary

then the more chances that it's going
to actually break the binary up and

destabilize it because if it doesn't
destabilize it the first time

it'll maybe destabilize it the second
time of the third time etc So

what you want in order to
stabilize one of these transients

binary is is essentially to
have a face to face where.

You don't have much chaos.

So even though chaos is responsible for
the formation of the binary in the first

place in terms of stabilization of
the binary subsequently you want to have.

Essentially only scattering

rather than chaos because otherwise
the you know the intruder will come in.

Because so much of face space is chaotic
the intruder will spend all of time

sort of glued to the binary.

And eventually will end
up destabilizing it and

it turns out that that for
these conditions to be met.

You need to have mass ratio olds of the
binary partners which are close to unity

and also moderate eccentricities
of the binary orbits.

So in a sense chaos is
is playing two roles.

It is initially responsible for

the formation of one of these
transients binary objects but.

It is also responsible for

the stabilization mechanism
due to intruders scattering.

Now in the last few years people
have discovered a very very

large binary is in the caper belled so
the binary is that I were talking about.

Up until this point are essentially in
this region so here is the mutual orbit

semi major axis as a fraction of the hill
sphere and here is the inclination or

the tilt the binary orbit makes through
a central to the plane of the solar system

and in in the last few years people
discovered sort of these humongous.

Binary objects which are very
very much larger than the other

population of binary as that was known
up until that point in the Cape about.

So our model it turns out here
is here is a mountain result of

a Monte Carlo simulation where we're
sure in slightly different way of.

Depicting the orbital elements here is
this is essentially inclination and

this is the result of our simulation so
we find by including dynamical friction

in addition to intrude a scattering in
our model that we can produce binary as

we each have seven major
axes which are comparable

to two goals actually observe so
I don't know if you can see here but

the green points correspond to
several of these very wide binary and

we get a distribution which is
certainly consistent with those.

So our model predicts the preference for

equal mass partners in Bahrain or
is it predicts

that predicts eccentricity is which are
comparable to two goals observed as well.

The same in major axes.

Recently many many binary
is of been discovered and

it turns out that in the three to
resonance in the solar system in fact

there is only one known wide binary that
has been found in the three to resonance.

And so the question arises is
this Kelso system capture model

is it consistent with new observations
namely that very few binary is or

in this case just one binary has been
discovered in the three to resonance.

So the three to resonance is
a resonance with Neptune and so

we can imagine that Neptune
is a periodic perturbation

which is operating on our
transience binary is.

And it also turns out that the that
orbits in the three to resonance

are on a fairly eccentric Helio
centric orbit so we have two

promises to play with one is that there
is the periodic kick from Neptune and

the other is the eccentricity
of the heliocentric orbit.

So here we look at the.

Sun.

Plus a binary This is a structure for
a space using something which is very

similar to the fastly happen off indicator
this is the make no chaos indicator and

here we have chaotic regions and
the stable regions a show on.

In blue.

It turns out that if you add
net two into the mix and

also change the eccentricity
of the heliocentric orbit.

Then the chaotic.

Region of face space.

Remember that the chaotic region
of face space is is the brown and

in particular the place where
binary has become captured

is that the interface between
the regular regions of the chaotic or

the far scattered regions the size
of this region shrinks appreciably.

So that means that there is
much less opportunity for

transients binary is to form.

If you allow.

Neptune to to to kick the hell
problem especially if you have.

Relatively large eccentricity.

So recently there have been and

body simulations reported
by this group in Japan.

So our calculations
are essentially just the.

Three three body problem the circular
restricted three body problem where

the whole problem in these
simulations something like

ninety thousand particles
were integrated and

the conclusion of this paper that
just came out last here is that

Kay also sisted capture emerges
naturally from the simulation.

So one of the things we were clearly
concerned about is that it's such a simple

model the circular restricted three body
problem is that going to survive intact

when you put in many many more particles
and it turns out it's the that it does.

OK so our conclusions are that.

As Joel said calles can be
both villainous and during So

in the formation of binary as in the belt
chaos essentially has a twin role.

It is responsible for the formation
of these transients binary is but

also it has a villainous role in the sense
that when it comes to stabilization

by scattering of a fourth body then
the amount of chaos which is present in

the problem is really a determinant for
the physical properties of the binary.

So the chaos assisted capture
model explains the roughly equal

mass ratio holes the size of
the orbits of the eccentricity is and

it also makes predictions specific
predictions about the ratio of

retrograde to pro-grade
binary is in the Kiper belt.

And this is one of the.

Testable predictions of the theory.

People are trying to determine
the sense there in the sense of knowing

of a mentor and whether you know the ratio
of binary to pro-grade orbits in.

We also predict that in
the three to resonance.

There will be fewer binarism in
the rest of the the character belt.

OK So observing campaigns
are underway to to really sort of

look at the detailed predictions of
our model so we'll see what happens.

OK I think in view of the time I'm
going to just skip to the end.

And not talk about this.

So.

So this is a very nice.

Quotation from Joe.

Or wait for George to finish but

I think I think this sums
up just Joe's curiosity.

Says he's.

Chairman thinks he does mathematics
the math department thinks it's astronomy.

There are even stronger ones
who believe in physics but

Joe himself liked to do
whatever he liked to do.

OK.

And these are some of the people who've
worked on these problems and students and

post-docs the observing campaign
to look at the ratio of

pro-grade to retrograde orbits is done in
collaboration with Daniel restaurant for

the observatory to Paris and

also theoreticians Annalyn Metra in
Belgium and Audrey compare in Belgium.

And also at this point I'd like to
thank her guy for inviting me and.

This is his birthday year.

And so I'll turn it over to the chair.